How can I generate a series of 2s and 3s such that the average of the generated values (so far) is as close to e as possible?

For example:

3:       avg=3      |2-e|    =0.282
3,2:     avg=2.5    |2.5-e|  =0.218
3,2,3:   avg=2.667  |2.667-e|=0.052
3,2,3,3: avg=2.75   |2.75-e| =0.032

Also, how can I quickly check if the nth index in the sequence is a 2 or 3?

How can I generate a series of 2s and 3s such that the average of the generated values (so far) is as close to $e$ as possible?

For example:

3:       avg=3      |2-e|    =0.282
3,2:     avg=2.5    |2.5-e|  =0.218
3,2,3:   avg=2.667  |2.667-e|=0.052
3,2,3,3: avg=2.75   |2.75-e| =0.032

Also, how can I quickly check if the $n$th index in the sequence is a 2 or 3?

How can I generate a series of 2s and 3s such that the average of the generated values (so far) is as close to e as possible?

For example:

2:       avg=2      |2-e|    =0.718
2,3:     avg=2.5    |2.5-e|  =0.218
2,3,3:   avg=2.667  |2.667-e|=0.052
2,3,3,3: avg=2.75   |2.75-e| =0.032

Also, how can I quickly check if the nth index in the sequence is a 2 or 3?

How can I generate a series of 2s and 3s such that the average of the generated values (so far) is as close to e as possible?

For example:

3:       avg=3      |2-e|    =0.282
3,2:     avg=2.5    |2.5-e|  =0.218
3,2,3:   avg=2.667  |2.667-e|=0.052
3,2,3,3: avg=2.75   |2.75-e| =0.032

Also, how can I quickly check if the nth index in the sequence is a 2 or 3?